Write \(\Pi _1\)  and \(\Pi _2\)  for the sets of prime numbers with residue 1
and 2 modulo 3, respectively. For any integer \(k\) , we denote by
\(\nu _p(k)\)  the \(p\) -adic valuation of \(k\) .
By [1]}, each \(p\in \Pi _2\)  is
irreducible in \(\mathcal {R}\) , and for \(p\in \Pi _1\) , there is an irreducible
element \(x_p \in \mathcal {R}\)  such that \(p=\mathcal {N}(x_p)\) . Furthermore,
we define
\(\mathcal {V}_n=\prod _{p \in \Pi _1} \lbrace 0 \le j_p \le \nu _p(3n+1) \rbrace ,\) 
